The likelihood function, by definition, is the probability density of
getting the data that we actually observed, as a function of the value
of . In other words, when we say that the likelihood function
peaks at , we mean that this value of is the
best fit to the data. In a Universe with , the data that
we actually observed are quite plausible. In a Universe with (where the likelihood is extremely small), the data that we
actually observed would have been quite unlikely to occur.

Now we're ready to state Bayes's theorem and apply it to
the situation at hand. Bayes's theorem says that the posterior
probability density is the product of the prior probability density
and the likelihood function (times a constant):

The constant of proportionality is chosen to make the posterior
probability density integrate to 1.

If we take the prior shown in Figure 1 and multiply it by the likelihood
function shown in Figure 2, we get the posterior probability density
shown in Figure 3. Over the relatively small range of 's where
the likelihood function was significantly different from zero (say
0.94-1.10 or so), the prior was essentially a constant plus that
narrow delta-function
spike at . That means that the posterior probability
will be just a bump shaped like the likelihood function plus
a delta-function spike as shown.

Figure 3:
My posterior probability density. The narrow spike is
a delta-function (infinitely high and narrow, with a finite area). For
the choice of prior shown in Fig. 1, 99.5% of the area of this curve
lies in the delta-function, and only 0.5% is in the broader bell curve.

You can work out what fraction of the total probability (that is,
of the integral of the probability density) lies under the delta-function
spike. For the prior shown in Figure 1, the answer turns out to be 99.5%.
In other words, the data changed me from a 15% believer in
to someone who things there are 200-to-1 odds in favor of .

It's perhaps of interest to know how much those numbers depend on the
details of the prior. Let's assume that a person's prior consists of
a delta-function spike at , plus some smooth function of
, and let's assume that the smooth bit can be taken to be
constant in the region where the likelihood is significant
(i.e., within a few standard deviations of ).
Then the prior can be characterized by two numbers: , the prior
probability
of inflation (the area under the narrow spike at ) and
, the ``background'' level in the vicinity.
For example, my prior as shown in Figure 1 is characterized by
=0.15 and
=0.01.

Figure 4
shows the posterior probability that as a function
of (the prior probability that ), for a few different
values of . I would argue that is about as large
as one would be likely to find.2

Figure 4:
Posterior probability that , as a function
of the prior probability. The parameter is the ``background
level'' of the prior in the vicinity of .

This shows that, quite generically, people who thought there was
any significant probability of before the recent data
rolled in should by now have very high assessments of the probability
that .

Unless, of course, one simply disbelieves
the data. The recent data come from several different experiments,
using different techniques and probing different aspects of the Universe,
and they all agree with each other. For that reason, I think that
the current data are quite robust. Anyway, my purpose at the moment
is not to convince you that you should believe the data, but
simply to show how the posterior probabilities come out if you do
believe the data.